Apply tilt to my rotating earth
Unless I am misunderstanding your question. You should be able to just tilt the earth and then rotate it setting keyframes how fast you want to do it. It should still follow the parent.

Dont do anything to the parent just set keyframes on the earth itself by rotating Z space. Not sure what the wobble is, check the graph editor to make sure nothing looks weird there. Maybe apply scale CTRL + A. Make sure you dont have to many keyframes that is making it look weird.